How to Translate 'There Exists Exactly One Happy Person' into Predicate Logic?

[tmt1]

How to translate "there exists exactly one happy person" into predicate logic?

I came up with $$ \exists x : happy(x) \implies \forall y: happy(y) \land y = x$$. But this is incorrect.

I also tried $$\exists x: happy(x) \land \forall y: happy(y) \land x = y$$. This is also incorrect.

The correct answer is :

$$ \exists x : (Happy(x) \land \forall y: Happy(y) \implies x = y))
$$

What is the error in my thinking?

 

[Ackbach]

$x$ or $y$ by themselves do not have truth values. You can't AND the $\text{happy}(y)$ with the $y$. You must predicate something about $y$. So, can you see that you must say $\exists x: \text{happy}(x)?$ This is why your first expression is wrong. An implication doesn't force the IF part to be true!

But why is the first part not enough? Because it gives you existence and not uniqueness. How can you get uniqueness? By forcing all other happy's to be equal to $x$. That's what the rest of the correct expression does.

Your second expression is wrong because it's (sort of) saying that there is a happy $x$, that everything else is happy, and that everything else is equal to $x$. That's WAY stronger than what you mean, isn't it? I would also add that there's probably a scope issue: I'm not sure $x$ is legally allowed to show up in the third expression unless you use parenthesis. Evgeny.Makarov can, I'm sure, give you more details on the correct syntax. He can also correct any faulty explanations on my part. ;)

 

[Evgeny.Makarov]

$x$ or $y$ by themselves do not have truth values. You can't AND the $\text{happy}(y)$ with the $y$.

I don't see this in the OP's post, at least after editing.

The first thing to decide is whether the scope of quantifiers is maximal or minimal. Tmt, please look in your textbook and describe the syntactic notations used there. In particular, whether $\forall x:A(x)\implies B(x)$ means $(\forall x:A(x))\implies B(x)$ or $\forall x:(A(x)\implies B(x))$.

 

[Ackbach]

I don't see this in the OP's post, at least after editing.

Ah, yes, I see now. Thanks!